Introduction to Complex Analysis 1:Differentiation
Seminar (13:01:50) ness: I want to introduce complex analysis, at a relatively basic level (13:02:24) ness: I will assume that you are fairly accustumed to the algebraic properties to the complex numbers and to real analysis (in a single variable) (13:02:52) ness: there might be slight allusions to multivariatle (2D) analysis but that's not essential (13:03:35) ness: I have created a list of figures (actually just two of them), I will refer to them throughout the seminar. (see above link) (13:04:33) ness: Let me say right from the beginning that I'm not going to strive for 100% rigour or completeness. Several important topics will be omitted (like multifunctions and riemann sourfaces) (13:05:03) ness: Now I think I should begin with a few historic remarks (13:05:34) ness: complex analysis has it's roots in 19th century. most of it's basic properties were developed by Cauchy (13:06:04) ness: many others have further contributed to the theory of course, important names are Riemann, Weierstrass, ... (13:06:25) ness: So much for the prelude, let's begin with the math (13:07:01) ness: oh and please tell me when I'm going too slow or too fast, it's hard to recognize these things when you can't see the audience (13:07:48) ness: We will first of all talk about convergence of sequences of complex numbers (13:08:10) ness: the definition is formally analogous to the real case (13:08:53) ness: so we define a sequence {a_n} to be a map N -> C from the positive integers to the complex numbers (13:09:51) ness: we could for example have a_n = n^2 + 4ni, this would yield the sequence (1+4i, 4+8i, 9+12i, ...) (13:10:39) ness: now we can define the limit of sequence as we would define it in a metric space (with the metric induced by the absolute value relation): (13:11:50) ness: we say that lim_{n->infty} a_n = c for c in C iff there exists for every real positive epsilon an n_0 such that for all n > n_0 |a_n-c| < epsilon (13:12:25) ness: throughout the seminar I will (from now on) assume that epsilon and delta are positive reals (13:13:39) ness: the first important point is that C can be proved to be complete, i.e. saying that the sequence {a_n} converges <=> {a_n} is cauchy (if you don't know what being cauchy means you can ignore that matter) (13:14:28) ness: as an example we can see that the sequence from above (a_n = n^2 + 4ni) cannot converge to any complex number, because it grows without bound (13:14:43) ness: there is one related notion, and that is absolute convergence (13:15:14) ness: we say that {a_n} converges absolutely if the sequence b_n = |a_n| converges (13:16:03) ness: the usual theorems known in the real case apply, so e.g. a sequence that converges absolutely also converges (in the normal sense), and the convergence tests for series also carry over (13:16:58) ness: we begin to see here a feature that can be found often in complex analysis: definitions that are analogous to the real case extend known concepts to the complex numbers, and many theorems carry over (13:17:15) ness: we will, however, see that complex analysis of course has a number of unique properties (13:17:57) ness: now that we have a feeling for how complex analysis is related to real analysis, let's begin with the more interesting stuff (13:18:50) ness: we must talk about convergence of functions, i.e. the notion that a function assumes a "limiting value" as its variable approaches a certain number (13:19:12) ness: in analogy to the real case, we define the following: (13:19:53) ness: let f:C -> C be a function (it could be defined on suitable subsets of C as well, but I don't go into these details here) (13:20:06) ness: further leg a, g be in C (13:21:02) ness: then we say that lim_{z->a} f(z) = g iff for every epsilon there exists a delta such that for all z with |z-a|0) (13:22:08) ness: we see that this is exactly the same definition that we would use in the real case, but, unlike the convergence of sequences, the convergence of functions is a much stronger statement than in the real case (13:22:31) ness: we will see a particularly striking example of this later (13:23:32) ness: of course we can also define limits of functions using sequences (like in the real case) yielding an equivalent construction (13:23:59) ness: for now, let's ignore other possibilites here and just define continuity: (13:24:51) ness: a function f:C->C is said to be continuous at a if lim_{z->a} f(z) = f(a), and we say that it is continuous in an open subset G of C if it is continuous in every a in G (13:25:20) ness: for example all polynomials and the absolute value function are continuous (13:26:12) ness: let's directly go on to one of the things that makes complex analysis unique, and that is differentiation (13:27:07) ness: if f:C -> C is a function, then we define the derivative of f at z by lim_{h->0} ((f(z+h)-f(z))/h if that limit exists (13:27:26) ness: so we see that the derivative is just a complex number (13:28:15) ness: and we can define the derivatve of f (the function), which is written Df or f' by f'(z) = the derivative of f at z (as defined above) (13:28:52) ness: all of these statements are formally analogous to the real case, but we will see shortly that complex differentiability is a much stronger statement than real differentiability (13:29:01) tkr: ! (13:29:09) ness: tkr: go ahead (13:29:13) tkr: what is h in there, real or complex, or does it matter? (13:29:23) ness: it is complex (13:29:27) tkr: ok (13:29:29) ness: and yes, it does matter (13:29:31) ness: a lot (13:30:20) tkr: ok. continue, please. :) (13:30:24) ness: for one thing (again nothing new), a function that is differentiable at z is also continuous (we say that a function is differentiable at z if it's derivative exists at z) (13:30:48) ness: before going into more formal matters, let's look at a geometric interpretation of differentiation (13:30:55) ness: consider figure 1 (13:31:18) ness: the left side shows a subset of the complex plane (13:31:45) ness: we think of an arbitrary complex number c, and small ("infinitesimal") vectors a and b starting from it (13:32:03) ness: with that we mean of course that a and b are arrows that connect c and some nearby numbers (13:32:21) ness: and now we can think of f as acting on c and the vectors (13:32:50) ness: that means that c becomes f© (13:33:00) ness: but what happens to a and b? (13:33:38) ness: well, of course the images of their tails are just f(z), and their heads are transformed according to whatever f does (13:34:09) ness: so we can say that in the w plane associated with a and b are their images (13:34:50) ness: in principle, f can rotate (twist) them and stretch (amplify) them separately by any amount (13:36:03) ness: but the definition of the derivative of f at z says that if f is differentiable, then the only thing that f "does" to the "infinitesimal" vectors a and b is to multiply both of them by *the same* complex number, namely f'© (13:37:03) ness: if you remember that multiplication by a complex number can be interpreted as a rotation followed by an amplification, you see that both a and be are rotated through the same angle, and amplified by the same factor (13:37:46) ness: we thus see that *if* f is differentiable at c (and f'© is not 0, that's a slight exception) then the angle between a and b is the same as the angle between their images (13:38:22) ness: thus a differentiable function preserves "infinitesimal shapes", or, speaking more formally, it preserves angles (13:39:02) ness: equipped with this interpretation, we can easily show that certain functions are not differentiable (13:39:30) ness: consider as an example the absolute value function as depicted in figure two (13:39:48) ness: it sends every complex number to a positive real (13:40:18) ness: (of course |x+iy|^2 = x^2 + y^2) (13:41:31) ness: so if we think again of our "infinitesimal" vectors a and b, it is clear that they are not generally rotated by the same amount (for they are parallel after f has acted, no matter where they started) (13:42:02) ness: this (somewhat sloppy) argument generalizes to all z and can be made perfectly formal (13:42:23) ness: so we see geometrically that |z| is nowhere differentiable (13:42:44) ness: this is, of course, very different from the real case, where |x| is differentiable almost everywhere! (13:43:17) ness: I'm stressing the geometric interpretation here because complex analysis is one of the subjects where geometric understanding and intuition can help a lot (13:44:23) ness: let's try to apply the same idea we used to show that the absolute value function is not differentiable to derive a general criterion (13:45:01) ness: we saw that, for a function to be differentiable, every "infinitesimal vector" must be rotated and amplified by the same amount (13:45:53) ness: when we look at the definitions, this is where it is important that the h in the definition of differentiability is a complex number (13:46:16) ness: let's repeat the definition: f'(z) = lim_{h->0} ((f(z+h)-f(z))/h (13:47:00) ness: what we have seen geometrically amounts to saying that we can let h approach zero along any direction, the limit must always come out the same (13:48:04) ness: we would for example let h go along the positive decreasing reals, i.e. we would take the difference quotient for h=0.1, h=0.01, and so on as approximations and look for the limit (13:48:31) ness: but someone else could go along the imaginary axis and try h=i*0.1, h=i*0.01 and so on (13:49:26) ness: and if f is differentiable, then we both necessarily will find the same answer! (this is just a "layman's" interpretation of the definition of a limit in complex analysis) (13:49:43) ness: let's now try to make this formal (13:50:36) ness: we let f:C -> C be a differentiable function where f(x+iy) = u(x, y) + iv(x, y). We think of u and v as R^2 -> R (13:51:30) ness: then we know that lim_{h -> 0, h in R} ((f(z+h)-f(z))/h = lim_{h->0, h in R} (f(z+ih)-f(z))/(ih) (13:52:00) ness: I explicitly wrote h in R here to stress that we want the old limit definition for real numbers here (13:52:17) ness: now we can substitute for f our sum of u+iv (13:53:24) ness: then the lhs of this equation yields lim_{h->0, h in R} 1/h * (u(x+h,y) + iv(x+h, y) - u(x, y) - iv(x, y)) (13:53:58) ness: and the rhs yields lim_{h->0, h in R} 1/(ih) (u(x, y+h) + iv(x, y+h) - u(x, y) - iv(x, y)) (13:54:02) ness: and they both must be equal (13:54:25) ness: but we know that if two complex numbers are two be equal, then their real and imaginary parts must be equal separately (13:54:31) ness: s/two/to (13:54:56) ness: so although this looks like one equation, it's actually two (real) equations (13:56:57) ness: if we split into real and imaginary parts, we obtain as one equation lim{h->0, h in R} 1/h * (u(x + h, y) - u(x, y)) = lim{h->0, h in R} 1/h * (v(x, y+h) - v(x, y) (13:57:14) ness: but these ordinary limits can be recognized as ordinary partials! (13:57:42) ness: so we see that u_x = v_y (the subscript meaning partial derivative) (13:58:09) ness: we can do the same thing for the imaginary part of the equations and obtain v_x = -u_y (13:58:12) ness: tada! (13:58:34) ness: from a geometric idea we have derived necessary (though not sufficient) conditions for a function to be complex differentiable (13:59:14) ness: now of course we are not the first to do so, credit is to be given to Cauchy and Riemann. For this reason the pair of differential equations is commonly known as the Cauchy-Riemann (CR) equations (13:59:29) ness: just one more point: (14:01:09) ness: if you apply the d/dx to u_x = v_y and d/dy to v_x = u_y, you see that u_xx = v_yx = v_xy = u_yy (if v_x and v_y are continous), so u is a so-called hamronic function (it's laplacian vanishes), and the same thing holds for v (14:01:26) ness: this is one source of a great many of applications of complex analysis (14:01:41) ness: that's it for today, I hope I haven't bored you too much (14:01:50) ness: are there any further questions? (14:02:05) tkr: well I think I need study the formulas, but some very basic questions (14:02:10) ness: sure (14:02:33) ness: it's mostly been ideas and concepts today (14:02:34) tkr: what is one trying to achieve by differentiating the complex function? ie. what makes you want to know the result? :) (14:03:03) tkr: differentiating a real you find the slope. thats simple. (14:03:41) ness: differentiating the complex function you find a number that is very analogous to the slope (with the interpretation given). Needham calls it an "amplitwist" (14:03:59) ness: of course complex functions don't have such a direct physical interpretation as real ones (14:04:38) ness: as one example, you can approximate complex functions using their derivative, as in the real case (the taylor formula also carries over) (14:05:10) tkr: ahaa. (14:05:14) ness: then there is the interesting point that in complex analysis any function that is once differentiable is actually infinitely many times differentiable (though we didn't prove that yet) (14:05:46) tkr: ok. planning on continuing with some of these topics next week? (14:06:03) ness: if there is interest, I would treat integration next week Category:Seminar Category:Complex Analysis